Does the spinning of a planet effects the attractions on other planets?

Question

Imagine we have two planets spinning very fast around their axis, lets say ones in every minute (both with the size of the Earth).

When they are at a distance will they get faster, slower or perhaps not all attracted to each other, or doesn't the spinning make any difference?

And is the direction of the spinning possibly of matter? So one is clockwise spinning and the other counter-clockwise spinning.

Of course the speed of spinning can be changed for better answers.

The composition of the planets can be simplified, so no water only a vast mass equally divided at a perfect sphere. So just a simple Earth.

Answer 1

If the two planets are rigid and spherically symmetric, the gravitational force between them is the same as if both masses were concentrated at their centres of mass. See the Shell Theorem. The strength and direction of this force does not change if the planets are rotated, because they are spherically symmetric. Therefore the attractive force between them does not depend on their spin, only the distance between them. Their spins have no effect on their motion.

If the planets are rigid but not spherically symmetric the force between them will vary as they are rotated. For example, if the planets are spherical but with an additional mass located at one point on the equator, then the attraction between these two extra masses will be strongest when they are closest; this attractive force tends to synchronise the spins of the planets, increasing the rotational speed of the slower while decreasing that of the faster.

Supposing that the planets orbit their overall centre of mass, the orbital and spin angular momenta will interact, while total angular momentum remains the same. The result is that after many rotations they orbit with the extra masses facing each other - a situation called tidal locking.

The same situation will arise if the planets are spherically symmetric but not perfectly rigid. The gravitational force on a planet is greatest closest to the other planet, and the resulting tidal force will cause a non-rigid planet to bulge towards the other. The movement of the bulge through the planetary material causes an overall loss of rotational energy, while also transferring angular momentum between the planets, slowing one planet while spinning up the other. See Tidal locking of a planet to a satellite.

Except in the case in which both planets are spherically symmetric, the force between them and their motion will be further complicated if the spin axes are not parallel and/or not perpendicular to the plane in which they orbit each other, or if the extra mass is not located symmetrically about this plane.

Answer 2

In a General relativistic (GR) description the rotation of an object will affect its gravitational attraction. The prominent effect for slow rotation would be "frame dragging". Apart form that rotation will deform the objects, this effect depends on rotation frequency and composition of the objects. When considering deformed stars things get very difficult.

To your question: a simple answer in a GR setting is not easy. If it where just one massive rotating body attracting a small test mass, then geodesics in the Kerr or Hartle-Thorne metric would give insight into particle motion around a rotating central object.

For two massive orbiting objects one would have to solve the general relativistic two body problem: searching for "Binary system of Kerr black holes" or in general for "relativistic Binaries" would be the right starting point.

For non compact objects (like a earth-like planet) "Two-body problem in post-Newtonian expansion" would be the right keyword. That being said those GR effects are rather minor for non compact systems. @sammy gerbil gave a nice answer on the situation in Newtonian gravity.

This answer is a summary of some keywords in this context because I can not give a "real" answer to the question. To my defense the question is rather board and a definite answer even to a more specific system in a GR setting could probably fill a research paper. GR or even post-Newtonian binaries are rather complicated and I know very little about specifics of the problem. If someone is an expert on the topic and has such an answer or has some good references I would be happy to up-vote such an answer!